Gelfand-Tsetlin Bases for Elliptic Quantum Groups

TL;DR

This paper classifies finite-dimensional irreducible representations of elliptic quantum groups and constructs Gelfand-Tsetlin bases, connecting algebraic and lattice model approaches with explicit formulas.

Contribution

It introduces a classification of representations via theta functions and constructs Gelfand-Tsetlin bases using Drinfeld generators and $L$-operators, linking different methods.

Findings

Classification theorem for irreducible representations

Explicit Gelfand-Tsetlin bases construction

Partition functions expressed via elliptic weight functions

Abstract

We study the level-0 representations of the elliptic quantum group $U_{q,p}(\widehat{\mathfrak{gl}}_N)$. We give a classification theorem of the finite-dimensional irreducible representations of $U_{q,p}(\widehat{\mathfrak{gl}}_N)$ in terms of the theta function analogue of the Drinfeld polynomial for the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$. We also construct the Gelfand-Tsetlin bases for the level-0 $U_{q,p}(\widehat{\mathfrak{gl}}_N)$-modules following the work by Nazarov-Tarasov for the Yangian $Y(\mathfrak{gl}_N)$-modules. This is a construction in terms of the Drinfeld generators. For the case of tensor product of the vector representations, we give another construction of the Gelfand-Tsetlin bases in terms of the $L$-operators and make a connection between the two constructions. We also compare them with those obtained by the first author by using the $\mathfrak{S}_n$-action realized by the elliptic dynamical $R$-matrix on the standard bases. As a byproduct, we obtain an explicit formula for the partition functions of the corresponding 2-dimensional square lattice model in terms of the elliptic weight functions of type $A_{N-1}$.